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Nonstandard analysis is a mathematical framework in which one extends the standard mathematical universe of standard numbers, standard sets, standard functions, etc. into a larger nonstandard universe of nonstandard numbers, nonstandard sets, nonstandard functions, etc., somewhat analogously to how one places the real numbers inside the complex numbers, or the rationals inside the reals. This nonstandard universe enjoys many of the same properties as the standard one; in particular, we have the transfer principle that asserts that any statement in the language of first order logic is true in the standard universe if and only if it is true in the nonstandard one. (For instance, because Fermat’s last theorem is known to be true for standard natural numbers, it is automatically true for nonstandard natural numbers as well.) However, the nonstandard universe also enjoys some additional useful properties that the standard one does not, most notably the countable saturation property, which is a property somewhat analogous to the completeness property of a metric space; much as metric completeness allows one to assert that the intersection of a countable family of nested closed balls is non-empty, countable saturation allows one to assert that the intersection of a countable family of nested satisfiable formulae is simultaneously satisfiable. (See this previous blog post for more on the analogy between the use of nonstandard analysis and the use of metric completions.) Furthermore, by viewing both the standard and nonstandard universes externally (placing them both inside a larger metatheory, such as a model of Zermelo-Frankel-Choice (ZFC) set theory; in some more advanced set-theoretic applications one may also wish to add some large cardinal axioms), one can place some useful additional definitions and constructions on these universes, such as defining the concept of an infinitesimal nonstandard number (a number which is smaller in magnitude than any positive standard number). The ability to rigorously manipulate infinitesimals is of course one of the most well-known advantages of working with nonstandard analysis.
To build a nonstandard universe from a standard one , the most common approach is to take an ultrapower of with respect to some non-principal ultrafilter over the natural numbers; see e.g. this blog post for details. Once one is comfortable with ultrafilters and ultrapowers, this becomes quite a simple and elegant construction, and greatly demystifies the nature of nonstandard analysis.
On the other hand, nonprincipal ultrafilters do have some unappealing features. The most notable one is that their very existence requires the axiom of choice (or more precisely, a weaker form of this axiom known as the boolean prime ideal theorem). Closely related to this is the fact that one cannot actually write down any explicit example of a nonprincipal ultrafilter, but must instead rely on nonconstructive tools such as Zorn’s lemma, the Hahn-Banach theorem, Tychonoff’s theorem, the Stone-Cech compactification, or the boolean prime ideal theorem to locate one. As such, ultrafilters definitely belong to the “infinitary” side of mathematics, and one may feel that it is inappropriate to use such tools for “finitary” mathematical applications, such as those which arise in hard analysis. From a more practical viewpoint, because of the presence of the infinitary ultrafilter, it can be quite difficult (though usually not impossible, with sufficient patience and effort) to take a finitary result proven via nonstandard analysis and coax an effective quantitative bound from it.
There is however a “cheap” version of nonstandard analysis which is less powerful than the full version, but is not as infinitary in that it is constructive (in the sense of not requiring any sort of choice-type axiom), and which can be translated into standard analysis somewhat more easily than a fully nonstandard argument; indeed, a cheap nonstandard argument can often be presented (by judicious use of asymptotic notation) in a way which is nearly indistinguishable from a standard one. It is obtained by replacing the nonprincipal ultrafilter in fully nonstandard analysis with the more classical Fréchet filter of cofinite subsets of the natural numbers, which is the filter that implicitly underlies the concept of the classical limit of a sequence when the underlying asymptotic parameter goes off to infinity. As such, “cheap nonstandard analysis” aligns very well with traditional mathematics, in which one often allows one’s objects to be parameterised by some external parameter such as , which is then allowed to approach some limit such as . The catch is that the Fréchet filter is merely a filter and not an ultrafilter, and as such some of the key features of fully nonstandard analysis are lost. Most notably, the law of the excluded middle does not transfer over perfectly from standard analysis to cheap nonstandard analysis; much as there exist bounded sequences of real numbers (such as ) which do not converge to a (classical) limit, there exist statements in cheap nonstandard analysis which are neither true nor false (at least without passing to a subsequence, see below). The loss of such a fundamental law of mathematical reasoning may seem like a major disadvantage for cheap nonstandard analysis, and it does indeed make cheap nonstandard analysis somewhat weaker than fully nonstandard analysis. But in some situations (particularly when one is reasoning in a “constructivist” or “intuitionistic” fashion, and in particular if one is avoiding too much reliance on set theory) it turns out that one can survive the loss of this law; and furthermore, the law of the excluded middle is still available for standard analysis, and so one can often proceed by working from time to time in the standard universe to temporarily take advantage of this law, and then transferring the results obtained there back to the cheap nonstandard universe once one no longer needs to invoke the law of the excluded middle. Furthermore, the law of the excluded middle can be recovered by adopting the freedom to pass to subsequences with regards to the asymptotic parameter ; this technique is already in widespread use in the analysis of partial differential equations, although it is generally referred to by names such as “the compactness method” rather than as “cheap nonstandard analysis”.
Below the fold, I would like to describe this cheap version of nonstandard analysis, which I think can serve as a pedagogical stepping stone towards fully nonstandard analysis, as it is formally similar to (though weaker than) fully nonstandard analysis, but on the other hand is closer in practice to standard analysis. As we shall see below, the relation between cheap nonstandard analysis and standard analysis is analogous in many ways to the relation between probabilistic reasoning and deterministic reasoning; it also resembles somewhat the preference in much of modern mathematics for viewing mathematical objects as belonging to families (or to categories) to be manipulated en masse, rather than treating each object individually. (For instance, nonstandard analysis can be used as a partial substitute for scheme theory in order to obtain uniformly quantitative results in algebraic geometry, as discussed for instance in this previous blog post.)
Many structures in mathematics are incomplete in one or more ways. For instance, the field of rationals or the reals are algebraically incomplete, because there are some non-trivial algebraic equations (such as in the case of the rationals, or in the case of the reals) which could potentially have solutions (because they do not imply a necessarily false statement, such as , just using the laws of algebra), but do not actually have solutions in the specified field.
Similarly, the rationals , when viewed now as a metric space rather than as a field, are also metrically incomplete, beause there exist sequences in the rationals (e.g. the decimal approximations of the irrational number ) which could potentially converge to a limit (because they form a Cauchy sequence), but do not actually converge in the specified metric space.
A third type of incompleteness is that of logical incompleteness, which applies now to formal theories rather than to fields or metric spaces. For instance, Zermelo-Frankel-Choice (ZFC) set theory is logically incomplete, because there exist statements (such as the consistency of ZFC) which could potentially be provable by the theory (because it does not lead to a contradiction, or at least so we believe, just from the axioms and deductive rules of the theory), but is not actually provable in this theory.
A fourth type of incompleteness, which is slightly less well known than the above three, is what I will call elementary incompleteness (and which model theorists call the failure of the countable saturation property). It applies to any structure that is describable by a first-order language, such as a field, a metric space, or a universe of sets. For instance, in the language of ordered real fields, the real line is elementarily incomplete, because there exists a sequence of statements (such as the statements for natural numbers ) in this language which are potentially simultaneously satisfiable (in the sense that any finite number of these statements can be satisfied by some real number ) but are not actually simultaneously satisfiable in this theory.
In each of these cases, though, it is possible to start with an incomplete structure and complete it to a much larger structure to eliminate the incompleteness. For instance, starting with an arbitrary field , one can take its algebraic completion (or algebraic closure) ; for instance, can be viewed as the algebraic completion of . This field is usually significantly larger than the original field , but contains as a subfield, and every element of can be described as the solution to some polynomial equation with coefficients in . Furthermore, is now algebraically complete (or algebraically closed): every polynomial equation in which is potentially satisfiable (in the sense that it does not lead to a contradiction such as from the laws of algebra), is actually satisfiable in .
Similarly, starting with an arbitrary metric space , one can take its metric completion ; for instance, can be viewed as the metric completion of . Again, the completion is usually much larger than the original metric space , but contains as a subspace, and every element of can be described as the limit of some Cauchy sequence in . Furthermore, is now a complete metric space: every sequence in which is potentially convergent (in the sense of being a Cauchy sequence), is now actually convegent in .
In a similar vein, we have the Gödel completeness theorem, which implies (among other things) that for any consistent first-order theory for a first-order language , there exists at least one completion of that theory , which is a consistent theory in which every sentence in which is potentially true in (because it does not lead to a contradiction in ) is actually true in . Indeed, the completeness theorem provides at least one model (or structure) of the consistent theory , and then the completion can be formed by interpreting every sentence in using to determine its truth value. Note, in contrast to the previous two examples, that the completion is usually not unique in any way; a theory can have multiple inequivalent models , giving rise to distinct completions of the same theory.
Finally, if one starts with an arbitrary structure , one can form an elementary completion of it, which is a significantly larger structure which contains as a substructure, and such that every element of is an elementary limit of a sequence of elements in (I will define this term shortly). Furthermore, is elementarily complete; any sequence of statements that are potentially simultaneously satisfiable in (in the sense that any finite number of statements in this collection are simultaneously satisfiable), will actually be simultaneously satisfiable. As we shall see, one can form such an elementary completion by taking an ultrapower of the original structure . If is the standard universe of all the standard objects one considers in mathematics, then its elementary completion is known as the nonstandard universe, and is the setting for nonstandard analysis.
As mentioned earlier, completion tends to make a space much larger and more complicated. If one algebraically completes a finite field, for instance, one necessarily obtains an infinite field as a consequence. If one metrically completes a countable metric space with no isolated points, such as , then one necessarily obtains an uncountable metric space (thanks to the Baire category theorem). If one takes a logical completion of a consistent first-order theory that can model true arithmetic, then this completion is no longer describable by a recursively enumerable schema of axioms, thanks to Gödel’s incompleteness theorem. And if one takes the elementary completion of a countable structure, such as the integers , then the resulting completion will necessarily be uncountable.
However, there are substantial benefits to working in the completed structure which can make it well worth the massive increase in size. For instance, by working in the algebraic completion of a field, one gains access to the full power of algebraic geometry. By working in the metric completion of a metric space, one gains access to powerful tools of real analysis, such as the Baire category theorem, the Heine-Borel theorem, and (in the case of Euclidean completions) the Bolzano-Weierstrass theorem. By working in a logically and elementarily completed theory (aka a saturated model) of a first-order theory, one gains access to the branch of model theory known as definability theory, which allows one to analyse the structure of definable sets in much the same way that algebraic geometry allows one to analyse the structure of algebraic sets. Finally, when working in an elementary completion of a structure, one gains a sequential compactness property, analogous to the Bolzano-Weierstrass theorem, which can be interpreted as the foundation for much of nonstandard analysis, as well as providing a unifying framework to describe various correspondence principles between finitary and infinitary mathematics.
In this post, I wish to expand upon these above points with regard to elementary completion, and to present nonstandard analysis as a completion of standard analysis in much the same way as, say, complex algebra is a completion of real algebra, or real metric geometry is a completion of rational metric geometry.